All results except (REF ) and (REF ) follow directly from [1]} and [2]} using multi-index notations.
By definition, \(\mathbb {E}_\theta [c_{\underline{i\hspace{-1.111pt}}\hspace{1.111pt}}c_{\underline{j\hspace{-1.111pt}}\hspace{1.111pt}}] = \eta _{\underline{i\hspace{-1.111pt}}\hspace{1.111pt}+\underline{j\hspace{-1.111pt}}\hspace{1.111pt}}(\theta )\) , hence using (), (REF ), and (REF ), we obtain (REF ).
For (REF ), observe that
\(\mathbb {E}_\theta [x^{\underline{i\hspace{-1.111pt}}\hspace{1.111pt}}] &= \idotsint _{\mathbb {R}^{d_x}} x_1^{\underline{i\hspace{-1.111pt}}\hspace{1.111pt}(1)} x_2^{\underline{i\hspace{-1.111pt}}\hspace{1.111pt}(2)} \cdots x_d^{\underline{i\hspace{-1.111pt}}\hspace{1.111pt}(d_x)} p(x,\theta ) dx_1\cdots dx_{d_x} \\&= \idotsint _{\mathbb {R}^{d_x-1}} x_1^{\underline{i\hspace{-1.111pt}}\hspace{1.111pt}(1)} x_2^{\underline{i\hspace{-1.111pt}}\hspace{1.111pt}(2)}\cdots x_{d_x-1}^{\underline{i\hspace{-1.111pt}}\hspace{1.111pt}(d_x-1)} \left[\int _{-\infty }^\infty x_{d_x}^{\underline{i\hspace{-1.111pt}}\hspace{1.111pt}(d_x)} p(x,\theta ) dx_{d_x}\right] dx_1\cdots dx_{d_x-1} \\&= \idotsint _{\mathbb {R}^{d_x-1}} x_1^{\underline{i\hspace{-1.111pt}}\hspace{1.111pt}(1)} x_2^{\underline{i\hspace{-1.111pt}}\hspace{1.111pt}(2)} \cdots x_{d_x-1}^{\underline{i\hspace{-1.111pt}}\hspace{1.111pt}(d_x-1)} \\& \times \left[ \left. \dfrac{x_{d_x}^{\underline{i\hspace{-1.111pt}}\hspace{1.111pt}(d_x)+1}}{\underline{i\hspace{-1.111pt}}\hspace{1.111pt}(d_x)+1} p(x,\theta ) \right|^\infty _{-\infty } -\int _{-\infty }^\infty \dfrac{x_{d_x}^{\underline{i\hspace{-1.111pt}}\hspace{1.111pt}(d_x)+1}}{\underline{i\hspace{-1.111pt}}\hspace{1.111pt}(d_x)+1} {p(x,\theta )}{x_{d_x}} dx_{d_x}\right] dx_1\cdots dx_{d_x-1}.\) 
